First, we need to understand the graph of the function \(f(x)=k^x\) (\(k\neq0\)).

As shown in the figure above we have 3 cases:
If \(k>1\), the graph of \(f\) is monotonically increasing, \(f(x)\geqslant1\) when \(x\geqslant0\), and \(f(x)<1\) when \(x<0\).
If \(k<1\), the graph of \(f\) is monotonically decreasing, \(f(x)\leqslant1\) when \(x\geqslant0\), and \(f(x)>1\) when \(x<0\).
If \(k=1\), then \(f(x)=1\).

Now consider 3 functions \(f(x)=a^x\), \(g(x)=b^x\) and \(h(x)=\frac{f(x)}{g(x)},\) where \(a>b>0.\) Then we have
\[h(x)=\frac{a^x}{b^x}=\left(\frac{a}{b}\right)^x.\]
As \(a>b>0\) and \(b\neq0\), we have \(\frac{a}{b}>1\).
Hence, \(h(x)=\left(\frac{a}{b}\right)^x\) is in the form \(k^x\) and \(k=\frac{a}{b}>1\), so the graph of \(h(x)\) is monotonically increasing.

When \(x>0\), we have \(h(x)=\frac{f(x)}{g(x)}>1\). Since \(g(x)>0\),
\[f(x)>g(x) \Rightarrow a^x>b^x.\]

When \(x<0\), we have \(h(x)=\frac{f(x)}{g(x)}<1\). Since \(g(x)>0\),
\[f(x)<g(x) \Rightarrow a^x<b^x.\]